Properties

Label 16.0.535...928.1
Degree $16$
Signature $[0, 8]$
Discriminant $5.358\times 10^{27}$
Root discriminant $54.08$
Ramified primes $2, 13, 17$
Class number $2448$ (GRH)
Class group $[2, 2, 6, 102]$ (GRH)
Galois group $C_{16} : C_2$ (as 16T22)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 85*x^14 + 2720*x^12 + 41531*x^10 + 311899*x^8 + 1087983*x^6 + 1861704*x^4 + 1531309*x^2 + 485537)
 
gp: K = bnfinit(x^16 + 85*x^14 + 2720*x^12 + 41531*x^10 + 311899*x^8 + 1087983*x^6 + 1861704*x^4 + 1531309*x^2 + 485537, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![485537, 0, 1531309, 0, 1861704, 0, 1087983, 0, 311899, 0, 41531, 0, 2720, 0, 85, 0, 1]);
 

\( x^{16} + 85 x^{14} + 2720 x^{12} + 41531 x^{10} + 311899 x^{8} + 1087983 x^{6} + 1861704 x^{4} + 1531309 x^{2} + 485537 \)

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 

Invariants

Degree:  $16$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(5357808174640126287142780928\)\(\medspace = 2^{16}\cdot 13^{4}\cdot 17^{15}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $54.08$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $2, 13, 17$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $8$
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{13} a^{10} - \frac{6}{13} a^{8} + \frac{3}{13} a^{6} - \frac{4}{13} a^{4} + \frac{3}{13} a^{2}$, $\frac{1}{13} a^{11} - \frac{6}{13} a^{9} + \frac{3}{13} a^{7} - \frac{4}{13} a^{5} + \frac{3}{13} a^{3}$, $\frac{1}{169} a^{12} - \frac{6}{169} a^{10} + \frac{55}{169} a^{8} + \frac{22}{169} a^{6} - \frac{49}{169} a^{4} + \frac{2}{13} a^{2}$, $\frac{1}{169} a^{13} - \frac{6}{169} a^{11} + \frac{55}{169} a^{9} + \frac{22}{169} a^{7} - \frac{49}{169} a^{5} + \frac{2}{13} a^{3}$, $\frac{1}{35827110046507} a^{14} - \frac{49207453587}{35827110046507} a^{12} - \frac{800979844207}{35827110046507} a^{10} + \frac{8509719033136}{35827110046507} a^{8} - \frac{11485647775495}{35827110046507} a^{6} - \frac{102281126757}{211994734003} a^{4} - \frac{6257481928}{16307287231} a^{2} + \frac{1471088465}{16307287231}$, $\frac{1}{35827110046507} a^{15} - \frac{49207453587}{35827110046507} a^{13} - \frac{800979844207}{35827110046507} a^{11} + \frac{8509719033136}{35827110046507} a^{9} - \frac{11485647775495}{35827110046507} a^{7} - \frac{102281126757}{211994734003} a^{5} - \frac{6257481928}{16307287231} a^{3} + \frac{1471088465}{16307287231} a$

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 

Class group and class number

$C_{2}\times C_{2}\times C_{6}\times C_{102}$, which has order $2448$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $7$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 3640.01221338 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{8}\cdot 3640.01221338 \cdot 2448}{2\sqrt{5357808174640126287142780928}}\approx 0.147852824429$ (assuming GRH)

Galois group

$OD_{32}$ (as 16T22):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_{16} : C_2$
Character table for $C_{16} : C_2$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, \(\Q(\zeta_{17})^+\)

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: Deg 32

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R $16$ $16$ $16$ $16$ R R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 
magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.8.8.4$x^{8} + 2 x^{7} + 2 x^{6} + 8 x^{3} + 48$$2$$4$$8$$C_8$$[2]^{4}$
2.8.8.4$x^{8} + 2 x^{7} + 2 x^{6} + 8 x^{3} + 48$$2$$4$$8$$C_8$$[2]^{4}$
$13$13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
17Data not computed